I have doubts in the following two questions :

  1. What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , $k,q=0,1,2,...$?

I know that the Fourier transform of $[x^k\varphi(x)]^{(q)}$ is $\xi^q\varphi^{(k)}(x)$. Also we know that Fourier transform takes differentiation to polynomial multiplication. Does a similar result hold for laplace Transform also or is it different ?

  1. How to prove the following assertion:

If an entire function $\varphi$ satisfies $$|\varphi(x+iy)|\le C\exp(a|x|^h+b|y|^{\gamma}), \:\:\:h\le\gamma$$ where $a\lt0$ and $C>0$, then $\varphi\in \mathcal{S}_{1/h}^{1-1/{\gamma}}$, where $\mathcal{S}_\alpha^\beta$ is the Gelfand-Shilov space of type $\mathcal{S}$ defined here Any help will be welcome. Thanks..

Reference: Generalized Functions, Volume 2, by I.M Gelfand and G.E. Shilov



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